# 3 modelling of physical systems

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<ul><li> 1. ME2142/ME2142E Feedback Control Systems 1 Modelling of Physical Systems The Transfer Function Modelling of Physical Systems The Transfer Function ME2142/ME2142E Feedback Control SystemsME2142/ME2142E Feedback Control Systems </li></ul>
<p> 2. ME2142/ME2142E Feedback Control Systems 2 Differential EquationsDifferential Equations Differential equation is linear if coefficients are constants or functions only of time t. Linear time-invariant system: if coefficients are constants. Linear time-varying system: if coefficients are functions of time. Differential equation is linear if coefficients are constants or functions only of time t. Linear time-invariant system: if coefficients are constants. Linear time-varying system: if coefficients are functions of time. PlantU YPlantU Y In the plant shown, the input u affects the response of the output y. In general, the dynamics of this response can be described by a differential equation of the form In the plant shown, the input u affects the response of the output y. In general, the dynamics of this response can be described by a differential equation of the form ub dt du b dt ud b dt ud bya dt dy a dt yd a dt yd a m m m m n n n n 01 1 101 1 1 3. ME2142/ME2142E Feedback Control Systems 3 Newtons Law f is applied force, n m is mass in Kg x is displacement in m. Newtons Law f is applied force, n m is mass in Kg x is displacement in m. m f x Mechanical Systems Translational SystemsMechanical Systems Translational Systems Mechanical Systems Fundamental LawMechanical Systems Fundamental Law Modelling of Physical Dynamic SystemsModelling of Physical Dynamic Systems xmmaf or 0 xmf xm DAlemberts Principle 4. ME2142/ME2142E Feedback Control Systems 4 T is applied torque, n-m J is moment of inertia in Kg-m2 is displacement in radians is the angular speed in rad/s T is applied torque, n-m J is moment of inertia in Kg-m2 is displacement in radians is the angular speed in rad/s J T Modelling of Physical Dynamic SystemsModelling of Physical Dynamic Systems Mechanical Systems Torsional SystemsMechanical Systems Torsional Systems JJT 0 JT J or 5. ME2142/ME2142E Feedback Control Systems 5 Rotational: T are external torques applied on the torsional spring, n-m G is torsional spring constant, n-m/rad Rotational: T are external torques applied on the torsional spring, n-m G is torsional spring constant, n-m/rad 1 2 Translational: f is tensile force in spring, n K is spring constant, n/m Translational: f is tensile force in spring, n K is spring constant, n/m f x1 x2 f K Modelling of Physical Dynamic SystemsModelling of Physical Dynamic Systems Mechanical Systems - springsMechanical Systems - springs )( 21 xxKf Important: Note directions and signs )( 21 GT 6. ME2142/ME2142E Feedback Control Systems 6 Translational: f is tensile force in dashpot, n b is coefficient of damping, n-s/m Translational: f is tensile force in dashpot, n b is coefficient of damping, n-s/m f x1x2 f . b . f x1x2 f . b . Modelling of Physical Dynamic SystemsModelling of Physical Dynamic Systems Mechanical Systems dampers or dashpotsMechanical Systems dampers or dashpots )( 21 xxbf Rotational: T is torque in torsional damper, n-m b is coefficient of torsional damping, n-m-s/rad Rotational: T is torque in torsional damper, n-m b is coefficient of torsional damping, n-m-s/rad 2 1 )( 21 bT 7. ME2142/ME2142E Feedback Control Systems 7 1 2 f x1 x2 f K Modelling of Physical Dynamic SystemsModelling of Physical Dynamic Systems Using superposition for linear systemsUsing superposition for linear systems Due to x1: 1Kxf 2Kxf Due to x2: )( 21 xxKf Due to both x1 and x2 : 2GT Due to :2 Due to : 1GT 1 )( 21 GTDue to both and :1 2 8. ME2142/ME2142E Feedback Control Systems 8 Translational damperTranslational damper f x1x2 f . b . f x1x2 f . b . Modelling of Physical Dynamic SystemsModelling of Physical Dynamic Systems Rotational damper:Rotational damper: 2 1 Using superposition for linear systemsUsing superposition for linear systems Due to : 1xbf 1x Due to :2x 2xbf )( 21 xxbf Due to both and :1x 2x 2bT Due to :2 Due to : 1bT 1 )( 21 bTDue to both and :1 2 9. ME2142/ME2142E Feedback Control Systems 9 Modelling of Physical Dynamic SystemsModelling of Physical Dynamic Systems ExampleExample Since m = 0, givesmaf 0 ds ff Since and Thus Or ybfd )( yxKfs 0)( ybyxK KxKyyb xy b K A Derive the differential equation relating the output displacement y to the input displacement x. Derive the differential equation relating the output displacement y to the input displacement x. Free-body diagram at point A, A fs fd Note: Direction of fs and fd shown assumes they are tensile. Note: Direction of fs and fd shown assumes they are tensile. 10. ME2142/ME2142E Feedback Control Systems 10 The transfer function of a linear time invariant system is defined as the ratio of the Laplace transform of the output (response) to the Laplace transform of the input (actuating signal), under the assumption that all initial conditions are zero. The transfer function of a linear time invariant system is defined as the ratio of the Laplace transform of the output (response) to the Laplace transform of the input (actuating signal), under the assumption that all initial conditions are zero. The Transfer FunctionThe Transfer Function Previous Example Assuming zero conditions and taking Laplace transforms of both sides we have Transfer Function This is a first-order system. Previous Example Assuming zero conditions and taking Laplace transforms of both sides we have Transfer Function This is a first-order system. KxKyyb )()()( sKXsKYsbsY Kbs K sX sY sG )( )( )( 11. ME2142/ME2142E Feedback Control Systems 11 Modelling of Physical Dynamic SystemsModelling of Physical Dynamic Systems ExampleExample Free-Body diagram givesmaf ods xmff m fs fd xo m K b xi xo )()()()()(2 sKXsbsXsKXsbsXsXms iiooo ooioi xmxxbxxK )()( iiooo KxxbKxxbxm Thus Or And Kbsms Kbs sX sX sG i o 2 )( )( )(Transfer Function . This is a second-order system. For the spring-mass-damper system shown on the right, derive the transfer function between the output xo and the input xi. For the spring-mass-damper system shown on the right, derive the transfer function between the output xo and the input xi. Note: fs and fd assumed to be tensile. 12. ME2142/ME2142E Feedback Control Systems 12 Capacitance Or Complex impedance Capacitance Or Complex impedance e q C Ceq dt de C dt dq i )(sECI )/(1 sCXc cIX sC IE 1 e i C Modelling of Physical Dynamic SystemsModelling of Physical Dynamic Systems Electrical ElementsElectrical Elements Resistance Units of R: ohms ( ) Resistance Units of R: ohms ( ) iRe R e i e i R Inductance Units of L: Henrys (H) Or Inductance Units of L: Henrys (H) Or dt di Le t te L i 0 d 1 )(sLIIXE L e i L IRE 13. ME2142/ME2142E Feedback Control Systems 13 Modelling of Physical Dynamic SystemsModelling of Physical Dynamic Systems Electrical Circuits- Kirchhoffs LawsElectrical Circuits- Kirchhoffs Laws Current Law: The sum of currents entering a node is equal to that leaving it. Current Law: The sum of currents entering a node is equal to that leaving it. 0i Voltage Law: The sum algebraic sum of voltage drops around a closed loop is zero. Voltage Law: The sum algebraic sum of voltage drops around a closed loop is zero. 0e 14. ME2142/ME2142E Feedback Control Systems 14 Modelling of Physical Dynamic SystemsModelling of Physical Dynamic Systems Electrical Circuits- ExamplesElectrical Circuits- Examples RC circuit: Derive the transfer function for the circuit shown, and giving This is a first-order transfer function. RC circuit: Derive the transfer function for the circuit shown, and giving This is a first-order transfer function. ci IXIRE co IXE )/(1 )/(1 sCR sC XR X E E c c i o 1 1 RCs ei i C R eo 15. ME2142/ME2142E Feedback Control Systems 15 Modelling of Physical Dynamic SystemsModelling of Physical Dynamic Systems Electrical Circuits- ExamplesElectrical Circuits- Examples RLC circuit: and giving This is a second-order transfer function. RLC circuit: and giving This is a second-order transfer function. cLi IXIXIRE co IXE )/(1 )/(1 sCsLR sC XXR X E E cL c i o 1 1 2 RCsLCs ei i C R eo L 16. ME2142/ME2142E Feedback Control Systems 16 Modelling of Physical Dynamic SystemsModelling of Physical Dynamic Systems Operational Amplifier Properties of an ideal Op AmpOperational Amplifier Properties of an ideal Op Amp Gain A is normally very large so that compared with other values, is assumed small, equal to zero. Gain A is normally very large so that compared with other values, is assumed small, equal to zero. )( 12 vvAvo )( 12 vv The input impedance of the Op Amp is usually very high (assumed infinity) so that the currents i1 and i2 are very small, assumed zero. The input impedance of the Op Amp is usually very high (assumed infinity) so that the currents i1 and i2 are very small, assumed zero. Two basic equation governing the operation of the Op Amp and Two basic equation governing the operation of the Op Amp and 0,0 21 ii2112 or0)( vvvv 17. ME2142/ME2142E Feedback Control Systems 17 Modelling of Physical Dynamic SystemsModelling of Physical Dynamic Systems Operational Amplifier ExampleOperational Amplifier Example For the Op Amp, assume i1=0 and vs=v+=0.For the Op Amp, assume i1=0 and vs=v+=0. - + vi i1 =0 voZi Zf ii if S Then orThen or0 fi ii 0 f o i i Z V Z V ThereforeTherefore i f i o Z Z sV sV )( )( i i f o V R Z V 18. ME2142/ME2142E Feedback Control Systems 18 Modelling of Physical Dynamic SystemsModelling of Physical Dynamic Systems Operational Amplifier ExampleOperational Amplifier Example - + vi i1 =0 voZi Zf ii if S i i f o V Z Z V For the following sC RZ ff 1 s K K CsRR R R Z V i p ii f i f i o 1V 19. ME2142/ME2142E Feedback Control Systems 19 Permanent Magnet DC Motor Driving a LoadPermanent Magnet DC Motor Driving a Load For the dc motor, the back emf is proportional to speed and is given by where is the voltage constant. The torque produced is proportional to armature current and is given by where is the torque constant. For the dc motor, the back emf is proportional to speed and is given by where is the voltage constant. The torque produced is proportional to armature current and is given by where is the torque constant. eK eK iKT t tK Relevant equations:Relevant equations: eaa K dt di LiRe iKT t b dt d JT e i Ra La eK J b T Note: By considering power in = power out, can show that Ke=KtNote: By considering power in = power out, can show that Ke=Kt 20. ME2142/ME2142E Feedback Control Systems 20 End </p>